Pseudo-spectral derivative of quadratic quasi-interpolant splines

More documents

Recommendations

Info

352 S. Remogna some results, obtained in [13], for the uniform case to the non uniform one. In Section 3 we construct the derivative of Q 2 f , called the pseudo-spectral derivative and the corresponding differentiation matrix D 2 . In Section 4 we propose the error analysis. Finally, in Section 5 we present some numerical results, giving comparisons with other known methods. We remark that the same technique could be used to estimate higher-order derivatives of f considering non uniform quasi-interpolant splines of higher degree, if the functionals defining them are known. 2. On a local discrete quadratic C 1 spline quasi-interpolant Let I =[a,b] be a bounded interval endowed with some partition ∆ k ={a=x 0 < x 1
Pseudo-spectral derivative 353 where f j = f(t j ) and h 2 j a j =− (h j−1 + h j )(h j−1 + 2h j + h j+1 ) , (3) h 2 j b j = 1+ (h j−1 + h j )(h j + h j+1 ) , h 2 j c j =− (h j + h j+1 )(h j−1 + 2h j + h j+1 ) . We can express (2) in the following form k+3 (4) Q 2 f = ∑ f j Ñ 2 j, j=1 where{Ñ 2 j }k+3 j=1 are the fundamental functions of Q 2, defined in [11] by Ñ 2 1 = N2 1 + a 2N 2 2 , (5) Ñ 2 j = c j−1 N 2 j−1 + b jN 2 j + a j+1N 2 j+1 , j = 2,...k+2, Ñ 2 k+3 = c k+2N 2 k+2 + N2 k+3 . 3. Pseudo-spectral derivative and differentiation matrix We define the pseudo-spectral derivative of f as the derivative of the quadratic quasiinterpolant spline Q 2 f . We denote it by Q ′ 2 f and we remark that it belongs toS 0 1 (∆ k), space of linear splines defined on ∆ k . Therefore, taking into account (2) and (4), we have: (6) Q ′ 2 f = ( k+3 ′ ∑ m j ( f)Nj) 2 k+3 = ∑ f j (Ñ 2 j) ′ . j=1 j=1 Since the derivative of the spline N 2 j [2] is (7) (N 2 j )′ (x)=2 ( N 1 j−1 (x) − N1 j (x) ) , j = 1,...,k+3, h j + h j−1 h j+1 + h j with N0 1(x)≡0, N1 k+3 (x)≡0 and{N1 j (x)}k+2 j=1 set of normalized linear B-splines span-
Page 1: Rend. Sem. Mat. Univ. Pol. Torino V
Page 5 and 6: Pseudo-spectral derivative 355 •
Page 7 and 8: Pseudo-spectral derivative 357 and,
Page 9 and 10: Pseudo-spectral derivative 359 Proo
Page 11 and 12: Pseudo-spectral derivative 361 We s

Pseudo-spectral derivative of quadratic quasi-interpolant splines

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?